Question: Solve for $x$ and $y$ using elimination. $\begin{align*}4x+2y &= -2 \\ 3x+5y &= 6\end{align*}$
Solution: We can eliminate $x$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-3$ and the bottom equation by $4$ $\begin{align*}-12x-6y &= 6\\ 12x+20y &= 24\end{align*}$ Add the top and bottom equations. $14y = 30$ Divide both sides by $14$ and reduce as necessary. $y = \dfrac{15}{7}$ Substitute $\dfrac{15}{7}$ for $y$ in the top equation. $4x+2( \dfrac{15}{7}) = -2$ $4x+\dfrac{30}{7} = -2$ $4x = -\dfrac{44}{7}$ $x = -\dfrac{11}{7}$ The solution is $\enspace x = -\dfrac{11}{7}, \enspace y = \dfrac{15}{7}$.